## 33 Citations

### Triposes, exact completions, and Hilbert's ε-operator

- Mathematics
- 2017

Abstract Triposes were introduced as presentations of toposes by J.M.E. Hyland, P.T. Johnstone and A.M. Pitts. They introduced a construction that, from a tripos P: Cop → Pos, produces an elementary…

### EXACT COMPLETION AND CONSTRUCTIVE THEORIES OF SETS

- MathematicsThe Journal of Symbolic Logic
- 2020

The theory of exact completions is used to study categorical properties of small setoids in Martin-Löf type theory and of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects.

### Topologies and free constructions

- Mathematics
- 2013

The standard presentation of topological spaces relies heavily on (naive) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are…

### Quasi-toposes as elementary quotient completions

- Mathematics
- 2021

The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category…

### Factorization systems induced by weak distributive laws

- Mathematics
- 2010

We relate weak distributive laws in SetMat to strictly associative (but not strictly unital) pseudoalgebras of the 2-monad (−) on Cat. The corresponding orthogonal factorization systems are…

### A NOTE ON EXACTNESS AND STABILITY IN HOMOTOPICAL ALGEBRA

- Mathematics
- 2001

Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of “homotopical exactness”, appearing for instance in the fibre or cofibre sequence of a…

### SLICING SITES AND SEMIREPLETE FACTORIZATION SYSTEMS

- Mathematics
- 2015

A factorization system (E;M) on a category A gives rise to the covariant category-valued pseudofunctor P of A sending each object to its slice category over M. This article characterizes the P so…

### Exactness and stability in homotopical algebra

- Mathematics
- 2000

Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of 'homotopical exactness', appearing for instance in the fibre or cofibre sequence of a…

## References

SHOWING 1-10 OF 33 REFERENCES

### Homotopy Structures for Algebras over a Monad

- MathematicsAppl. Categorical Struct.
- 1999

The interest is given by the possibility of lifting the ‘homotopy operations’ (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to AT, just by verifying the consistency between these operations and λ : T P → P T.

### Homotopical algebra in homotopical categories

- MathematicsAppl. Categorical Struct.
- 1994

A hierarchy of notions ofh-,h1-, ...h4-categories is introduced, through progressive enrichment of thevertical structure of homotopies, so that the strongest notion, h4-category, is a sort of relaxed 2-category.

### Stable Homotopy Monomorphisms

- MathematicsAppl. Categorical Struct.
- 1997

The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather'shomotopy pullback, and shows that these modified homotology monomorphisms are exactly those homotope pullback stable, hence the terminology “stable” homotopies.

### Categorically Algebraic Foundations for Homotopical Algebra

- MathematicsAppl. Categorical Struct.
- 1997

This structure, if somewhat heavy, has the interest of beingategorically algebraic, i.e., based on operations onfunctors, and can be naturally lifted from a category A to its categories of diagrams AS and its slice categories A\X,A/X.

### Homotopy is not concrete

- Mathematics
- 1970

not null-homotopic such that [Z,X] [Z, f ] → [Z, Y ] is constant. For any X ′ ⊂ X with fewer than κ cells there exists X ′ → Z → X ′ = 1X′ , from which we may conclude that [X ′, X] [X ′, f ] → [X ′,…

### Pull-Backs in Homotopy Theory

- MathematicsCanadian Journal of Mathematics
- 1976

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no…